Rational Root Theorem · Explained · Examples · Practice

In this lesson, you will see the rational root theorem, or the rational zero theorem, and how to use it with guided examples.

After that, there are test style practice questions for you!

If you’re comfortable with what rational numbers and roots are, then you can skip ahead to the theorem itself.

Keep this table of factors and multiples handy while you’re working through the examples and practice questions!

Contents

What is a Rational Number?
What is a Root?
The Rational Root Theorem (Rational Zero Theorem)
How to Use It
Examples
Why Use It?
What is the Integral Root Theorem?
Practice
To Sum Up (Pun Intended!)

What is a Rational Number?

Rational and Irrational Numbers

A ratio is a fractional relationship between two quantities!

A ratio-nal number is a quantity that can be written as a fraction.

What is a Root?

A root is a value for x that makes the function equal to zero. It is also called a solution.

This function f(x) is a polynomial of order 3. The highest power of x is 3.

Roots Explained

The graph of the function crosses the x-axis at points x=-1 and x=1. These are the x-intercepts because at these points on the function, y=0.

So the roots, or solutions, of function f(x) are x=-1 and x=1.

In the case above, the root x=-1 is repeated because it is also a local maximum.

The Rational Root Theorem (Rational Zero Theorem)

Also known as the rational zero theorem, the rational root theorem is a powerful mathematical tool used to find all possible rational roots of a polynomial equation of the order 3 and above.

Polynomial Order and Coefficients

The rational root theorem says that if there are rational roots, they will be one of the following:

Rational Root Theorem

This means that the roots of the equation are one of the combinations of ± the factors of the constant-coefficient a0 divided by ± the factors of the n^coefficient an.

For the theorem to work, the coefficients an, an-1, an-2, etc., must be integers – whole numbers.

You’ll see how it works in the next section.

Only use the rational root theorem when the coefficients are small, and as a last resort or when you’re told to. It is less efficient than some other steps you could take first.

First, try factoring as much as possible, it may turn out you don’t need to use this method if you can simplify a polynomial of a higher order, like x3 or x4, by reducing it to a combination of lower-order equation(s), for example:

Factoring x^5+7x^3-18x to x(x2+9)(x2-2)

In that case, there is no need to use a theorem to solve the equation – factoring did it for you!

How to Use the Rational Root Theorem

First, identify the coefficients you will need to use.

Identifying Coefficients

Let’s say that you’ve been asked to list all the possible rational roots for the last equation in the diagram above:

3x3-4x2-17x+6=0

Use the rational root theorem.

The coefficients are an=3 and a0=6.

The factors of 3 are 1 and 3, and the factors of 6 are 1, 2, 3, and 6. So the possible roots are:

Possible Roots for a_n=3 a_0=6

Put each number into the function as x to test if it’s a root, like this.

Roots Test for 3x3-4x2-17x+6

If f(x)=0, then the value is a rational root, so you can see that putting x={-2, 13, 3} gives 0.

So the polynomial has 3 rational roots, x=-2, x=13, and x=3.

A polynomial always has the same number of roots as its order.

Reminder: a polynomial’s order is the highest power of the variable – in most cases, the variable is x or t.

So a polynomial of order 5 has 5 roots. That doesn’t mean they are all rational though.

A polynomial can have no rational, or real, roots. This just means that it’s roots are complex, and involve some irrational operator, like a radical, e, π, etc.

If the theorem finds no zeros, the polynomial has no rational roots.

Examples

a) List the possible rational roots for the function

f(x) = x4 + 2x3 – 7x2 – 8x + 12

b) Test each possible rational root in the function to confirm which are solutions to f(x)=0.

c) Use the confirmed rational roots to factorize the polynomial.


a) To find the possible rational roots, use the theorem: ± the factors of the constant-coefficient 12 divided by the factors of the x4-coefficient 1.

Solution to Example 1 part 1

b) For each possible rational root, replace x with the value and evaluate the function.

Solution 1 2

c) The confirmed roots are the ones that made the function equal to zero.

Solution 1 3

In this last example, did you notice that when we found roots, like negative 3, the 3 is positive when writing out the factorization? The sign changed. Why is that?

It’s because each bracket in the factorization must equal zero when x is that root. A root of -3 will make zero when 3 is added to it.

At the point in the graph where x=-3, (x+3) is actually (-3+3)=0.

The whole function multiplied by 0 is 0!

a) List the possible rational roots for the function

f(x) = 2x3 + 7x2 – 37x – 42

Here are the factors of 42 for your reference.

b) Test each possible rational root in the function to confirm which are solutions to f(x)=0.

c) Use the confirmed rational roots to factorize the polynomial.


a) To find the possible rational roots, use the theorem: ± the factors of the constant-coefficient, 42, divided by the factors of the x3-coefficient, 2.

Solution to Example 2 part 1

b) For each possible rational root, replace x with the value and evaluate the function.

Solution to Example 2 part 2

c) The confirmed roots are the ones that made the function equal to zero.

Solution to Example 2 part 3

Why Use It?

In advanced mathematics, there will be times when you must find the solutions or roots of a polynomial for a practical, or theoretical purpose.

You may want to find the point of greatest flow while on the design team for a rollercoaster at a theme park, which can be modeled using a polynomial equation, or perhaps you’ll need to theoretically test the aerodynamics of a curved edge before building an expensive prototype.

What is the Integral Root Theorem?

There is a special case of the rational root theorem, where the coefficient an=1, called the integral root theorem.

Practice

Use the rational root theorem to list all possible rational roots for the equation x3+2x-9=0.


Practice Solution 1

Use the rational root theorem to factorize the following polynomial function:

f(x)=2x4-11x3+4x2+14x-3

Only factorize as much as you can using the theorem.


First, use the theorem to find the possible rational roots.

Practice Solution 2

Then plug those values into f(x) as x to see if they are roots.

Practice Solution 2 part 2

Finally, use the known roots to factorize as much as you can.

Practice Solution 2 part 3

The focus of this lesson is using the theorem, but for those of you that took it to the next level and used polynomial long or synthetic division, the quadratic is (x2-5x+1).

How did you get on with the questions? Were there any parts you weren’t confident with? Any questions you have can be asked in the comments at the end of the lesson.

To Sum Up (Pun Intended!)

Rational numbers can be written as a fraction of whole numbers, or as a decimal number that either has an end, like 0.65, or a repeating pattern like 0.16161616…

The roots of a polynomial f(x) are values of x that solve the equation f(x)=0.

As the name suggests, a rational root is the combination of a rational number with a root.

The rational root theorem, which is also called the rational zero theorem, says that any rational roots of the polynomial must be one of the following:

Rational Root Theorem Full

Don’t forget your handy quick reference guide for factors.

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